Nuprl Lemma : intuitionistic-pigeonhole
∀A,B:ℕ ⟶ ℙ.
((∀s:StrictInc. ⇃(∃n:ℕ. A[s n])) ⇒ (∀s:StrictInc. ⇃(∃n:ℕ. B[s n])) ⇒ (∀s:StrictInc. ⇃(∃n:ℕ. (A[s n] ∧ B[s n]))))
Proof
Definitions occuring in Statement :
strict-inc: StrictInc,
quotient: x,y:A//B[x; y],
nat: ℕ,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s],
strict-inc: StrictInc,
prop: ℙ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
compose: f o g,
nat: ℕ,
subtype_rel: A ⊆r B,
guard: {T},
cand: A c∧ B,
and: P ∧ Q
Lemmas referenced :
two-implies-quotient-true,
and_wf,
implies-quotient-true,
canonicalizable-nat-to-nat,
less_than_wf,
int_seg_wf,
canonicalizable-set,
canonicalizable_wf,
trivial-quotient-true,
axiom-choice-quot,
compose-strict-inc,
equiv_rel_true,
true_wf,
exists_wf,
quotient_wf,
all_wf,
strict-inc_wf,
nat_wf,
unary-almost-full-has-strict-inc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_functionElimination,
because_Cache,
isectElimination,
independent_isectElimination,
functionEquality,
cumulativity,
universeEquality,
natural_numberEquality,
productElimination,
dependent_pairFormation,
independent_pairFormation
Latex:
\mforall{}A,B:\mBbbN{} {}\mrightarrow{} \mBbbP{}.
((\mforall{}s:StrictInc. \00D9(\mexists{}n:\mBbbN{}. A[s n]))
{}\mRightarrow{} (\mforall{}s:StrictInc. \00D9(\mexists{}n:\mBbbN{}. B[s n]))
{}\mRightarrow{} (\mforall{}s:StrictInc. \00D9(\mexists{}n:\mBbbN{}. (A[s n] \mwedge{} B[s n]))))
Date html generated:
2016_05_14-PM-09_48_45
Last ObjectModification:
2016_01_06-PM-09_29_05
Theory : continuity
Home
Index